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Seidel adjacency matrix
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, the Seidel adjacency matrix of a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">simple undirected graph</a> <i>G</i> is a <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a> with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices. It is also called the Seidel matrix or – its original name – the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the <a href="/facts/Adjacency_matrix/jYXNhYGs">adjacency matrix</a> of <i>G</i> from the adjacency matrix of the <a href="/facts/Complement_graph/IdHUPFr3">complement</a> of <i>G</i>. </p><p>The <a href="/facts/Multiset/bPFJGMDo">multiset</a> of <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of this matrix is called the Seidel spectrum. </p><p>The Seidel matrix was introduced by <a href="/facts/J._H._van_Lint/JMDwvfkS">J. H. van Lint</a> and Johan Jacob Seidel [de; nl] in 1966 and extensively exploited by Seidel and coauthors. </p><p>The Seidel matrix of <i>G</i> is also the adjacency matrix of a <a href="/facts/Signed_graph/WAErBGEB">signed complete graph</a> <i>KG</i> in which the edges of <i>G</i> are negative and the edges not in <i>G</i> are positive. It is also the adjacency matrix of the <a href="/facts/Two-graph/edBYfUma">two-graph</a> associated with <i>G</i> and <i>KG</i>. </p><p>The eigenvalue properties of the Seidel matrix are valuable in the study of <a href="/facts/Strongly_regular_graph/0GbdLwSd">strongly regular graphs</a>. </p> <ul><li>van Lint, J. H., and Seidel, J. J. (1966), Equilateral point sets in elliptic geometry. <i>Indagationes Mathematicae</i>, vol. 28 (= <i>Proc. Kon. Ned. Aka. Wet. Ser. A</i>, vol. 69), pp. 335–348.</li> <li>Seidel, J. J. (1976), A survey of two-graphs. In: Colloquio Internazionale sulle Teorie Combinatorie (Proceedings, Rome, 1973), vol. I, pp. 481–511. Atti dei Convegni Lincei, No. 17. Accademia Nazionale dei Lincei, Rome.</li> <li>Seidel, J. J. (1991), ed. <a href="/facts/Derek_Corneil/IlyG5ET3">D.G. Corneil</a> and R. Mathon, Geometry and Combinatorics: Selected Works of J. J. Seidel. Boston: Academic Press. Many of the articles involve the Seidel matrix.</li> <li>Seidel, J. J. (1968), Strongly Regular Graphs with (−1,1,0) Adjacency Matrix Having Eigenvalue 3. <i>Linear Algebra and its Applications</i> 1, 281–298.</li></ul>